Complex singular spectrum analysis of earth orientation time series

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Geodätisches Institut

Complex Singular Spectrum Analysis

of Earth Orientation Time Series

Masterarbeit im Studiengang

Geodäsie und Geoinformatik

an der Universität Stuttgart

Yang LI

Stuttgart, Juli 2018

Betreuer: Prof. Dr.-Ing. Nico Sneeuw Universität Stuttgart

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Abstract

The separation of geodetic time series into a sum of a small number of independent and interpretable components is of great importance. Approaches such as Singular Spectrum Analysis (SSA) and Multi-channel Singular Spectrum Analysis (MSSA) have shown their advantage in this field.

The aim of this thesis is to develop theoretical aspects of Complex Singular Spectrum Analysis (CSSA) technique and demonstrate that CSSA can be considered as a powerful method of time series analysis. CSSA is a non-parametric method which extends SSA into bivariate time series analysis technique with complex values.

Using EOP time series in the time period spanning from 1960 to 2009, Chandler wobble (CW), annual wobble (AW) as well as a non-linear trend are successfully separated in bothxandydirection by SSA and MSSA. This paper provides evidence for CSSA when performing tasks of EOP time series decomposition. CSSA functions quasi-equivalent with SSA and MSSA of polar motion time series analysis.

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List of Abbreviations

AW AnnualWobble

CSSA ComplexSingularSpectrumAnalysis

CW ChandlerWobble

EOF EmpiricalOrthogonalFunctions

EOP EarthOrientationParameters

ICA IndependentComponentAnalysis

LOD LengthOfDay

MSSA MultivariateSingularSpectrumAnalysis

RMS RootMeanSquare

PC PrincipalComponent

PCA PrincipalComponentAnalysis

SSA SingularSpectrumAnalysis

SVD SingularValueDecomposition

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Chapter 1

Introduction

1.1 Motivation

The purpose of time series analysis is to determine some of the system’s key properties which can then help understand and predict the system’s future behavior by quantifying certain features of the time series (Ghil et al., 2002). In the geodesy, data-driven methods, such as Empirical Orthogonal Function (EOF), Principal Component Analysis (PCA) and Independent Component Analysis (ICA), are used to smooth or model the geodetic time series (Dong et al. (2006); Rangelova et al. (2007); Forootan and Kusche (2012)). Singular Spectrum Analysis (SSA) in its turn is a generalization of PCA and Multi-channel Singu-lar Spectrum Analysis (MSSA) is known as an Extended EOF analysis (Zotov and Shum, 2010).

Over the last two decades, SSA and MSSA have proven their usefulness in the temporal and spatial-temporal analysis of short and noisy time series in several fields of the geo-sciences and of other disciplines (Ghil et al., 2002).

The starting point of SSA and MSSA is to embed a time series in a vector space, to repre-sent it as a trajectory matrix that generate the time series numerically (Ghil et al., 2002). But SSA can be just used of uni-variate time series analysis. And trajectory matrix for MSSA is too complex. So introduce CSSA whose trajectory matrix is simple and can an-alyze multivariate time series.

It is worth mentioning these two methods have been shown to be successful in decompo-sition of polar motion time series into a sum of trend, Chandler wobble, annual wobble and residual.

This thesis is trying to find out a new method, CSSA, to analyze bivariate time series by modelling them as a single-channel complex valued time series. It is important to note that CSSA is used for the first time to analyze polar motion observations. This thesis want to prove that CSSA performs well in multivariate geodetic time series.

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1.2 The Earth Orientation Parameters (EOP) time series

The Earth Orientation Parameters (EOP) time series consists of polar motion inx direc-tionxpandydirectionyp, Universal Time (UT1), Length of Day (LOD), and celestial pole

offsets of precession and nutation. This thesis will analyze polar motion observations from the year1960to2009both inxandydirection in time domain. It is shown as fol-lowing:

FIGURE1.1: Polar motion from 1960 to 2009 inxandydirection

Polar motion (or wobble) is the moving of the Earth’s rotation axis with respect to the con-ventional Earth-fixed reference system. Polar motion plays an important role in our un-derstanding of the processes that drive changes in the Earth’s rate of rotation to change, solid Earth, atmosphere, and oceans (Malkin and Miller, 2010; Smylie et al., 2015; Höpfner, 2003). It has three major components:

1. A free oscillation called Chandler wobble with period about435days.

2. Annual oscillation (annual wobble) forced by the seasonal displacement of air and water masses, beating which each other, give the characteristic pulsating shape of the motion (Vondrák and Richter, 2004).

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The annual oscillation proceeds in the same direction as Chandler wobble with nearly a constant amplitude of about 0.1 [arcsec]. The mean pole has an irregular drift in the direction to the80◦W (Vondrák and Richter, 2004).

This secular motion is mainly due to glacial isostatic adjustment in Canada and Scandi-navia, but sea-level changes, large-scale tectonic movements, mass shifts in the Earth’s interior and polar ice melting may also contribute to this trend. It is also called polar wander (Torge and Müller, 2012).

Meanwhile there are also some other free motions in addition to Chandler wobble, due to misalignments of rotational axis and figure axes related to the flattened Earth’s mantle, and inner and outer core (Torge and Müller, 2012).

These components for free motions will be seen together as residual. There has been some research about the decomposition. These decompositions for polar motion inxdirection is shown as following:

FIGURE1.2: Polar motion decomposition inxdirection.

Source: plot provided by the former Central Bureau. (Created: 1 Jan 2001). https://www.iers.org/IERS/EN/Science/EarthRotation/Xpole.html?nn=12932.

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Decomposition inydirection is shown as following:

FIGURE1.3: Polar motion decomposition inydirection.

Source: plot provided by the former Central Bureau. (Created: 1 Jan 2001). https://www.iers.org/IERS/EN/Science/EarthRotation/Ypole.html?nn=12932.

1.3 Window length choice of window length on polar motion

series

As mentioned in section 2, the window lengthLis the only parameter in the decomposi-tion algorithms of embedding step. So it is really important to choose the proper window length. The selection depends on the objective of the study and on preliminarily infor-mation about the time series.

This selection of small values ofLhas the advantage of increasing the confidence in the results when the objective of the analysis present high frequencies (Oropeza and Sacchi, 2011a). Other authors have said thatLhas to be sufficiently large so that the main behav-ior of the time series to analyze is content in each lagged vector (Golyandina et al., 2001) and theoretical results tell us that Lshould only be large enough but smaller thanN/2

(Hassani, 2007).

Polar motion time series has two periodic oscillations components with an integer pe-riod, for CW with periods of about435and AW with365days (Höpfner, 2003). Then it is advisable to take the window length proportional to the periods in order to get better

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separability of these periodic components (Hassani, 2007). For polar motion time series analysis, in order to separate the oscillations components (CW and AW), also meanwhile the trend, the most appropriate window lengthLequals12to13years both inxdirection and inydirection (Cui, 2015). As we know from the EOP data set, there are20 observa-tions per year, so the size of the window length in this thesis will be chosen as240.

1.4 Structure of thesis

Chapter1 gives motivation of the thesis with the study history and the application of the methods and introduction of Earth Orientation Parameters (EOP) time series. Then it gives the definition and content of polar motion which is the part we will analyze. What’s more, the explanation and pre-knowledge of polar motion are really helpful for the analysis.

The rest of this thesis is organized as follows. In Chapter 2, the introduction for SSA is given first with definition, study history and the usefulness. Then the algorithm of SSA is explained in mathematical formulas. What’ more, applying SSA of polar motion time series inxdirection and then inydirection. At last, we will do some discussions for the results.

On the other hand, Chapter 3 introduces the expansion of SSA to multiple dimensions called MSSA. Background and algorithm of MSSA are given. And it shows the results for analyzing the polar motion time series both inxandydirection with MSSA. Finally, it gives the discussions on analysis.

Chapter 4 shows background and algorithm of Complex Singular Spectrum Analysis (CSSA) which is expansion of SSA in the complex field. Then it gives results of separating main components from polar motion time series with CSSA and does some discussions on the results got by different generations.

Chapter 5 compares the methods SSA, MSSA with CSSA and show the difference in fig-ures. Then conclusion can be given by the comparison of this three methods. At last, it gives the conclusion and outlook of the whole thesis.

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Chapter 2

Singular Spectrum Analysis

This chapter gives the background for Singular Spectrum Analysis (SSA) (Section 2.1) with its history and application, the basic algorithms for SSA with further details on the four main steps (Section 2.2), as well as the emphasizing in its application for de-composing polar motion time series that demonstrate its effectiveness in extracting main components (Section 2.3) and analyzing with amplitude and frequency got by fitting.

2.1 Background

In time series analysis, SSA is a nonparametric spectral estimation method. SSA can be an aid in the decomposition of time series into a sum of a small number of independent and interpretable components such as a slowly varying trend, oscillatory components and a structureless noise, each having a meaningful interpretation (Hassani, 2007). The name "Singular Spectrum Analysis" relates to the spectrum of eigenvalues in a Singular Value Decomposition (SVD) of a covariance matrix, and not directly to a frequency do-main decomposition.

SSA are called the Caterpillar method (Golyandina et al., 2001) and Cadzow filtering (Trickett, 2008) in different fields. All this names of methods arise from different fields, but their methods are the equivalent. For instance, the Caterpillar method arises from time series analysis (Nekrutkin, 1996) and the Cadzow method was proposed as a gen-eral framework for signal enhancement, denoising images (Cadzow, 1988) and applica-tion for random noise attenuaapplica-tion (Trickett, 2008).

Areas where SSA can be applied are very broad: climatology, marine science, geophysics, engineering, image processing, medicine, econometrics (Golyandina et al., 2001). Hence different modifications of SSA have been proposed and different methodologies of SSA are used in practical applications such as trend extraction, periodicity detection, seasonal adjustment, smoothing, noise reduction (Golyandina et al., 2001). It is an important fea-ture of SSA that the trends need not be linear and that the oscillations can be amplitude and phase modulated (Ghil et al., 2002).

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The application of SSA consists of two complementary stages with four main steps – em-bedding, Singular Value Decomposition(SVD), grouping and anti-diagonal averaging. In the next section, the methodology will be explained in detail.

2.2 Methodology

Consider the real-valued time series Y = (x1, x2, x3,· · · , xN) of length N. The main

purpose of SSA is to decompose the original time series into a sum of independent and interpretable components such as a slowly varying trend, periodic or quasi-periodic com-ponents and a structureless noise (Hassani, 2007).

This is followed by two complementary stages: decomposition and reconstruction. In the first stage, we decompose the time series in two steps. The first step is embedding original time seriesY with lagged vectorXiinto the trajectory matrixX. The second step

singular value decomposition represents trajectory matrix Xas a sum of bi-orthogonal elementary matricesXi whose rank is one. They will turn polar motion time series into

the trend, Chandler wobble, annual wobble and residual according to their singular val-ues. Reconstruction is the second stage. This stage includes two separate steps: group-ing and anti-diagonal averaggroup-ing. We reconstruct the original time series by groupgroup-ing to make subgroups of the decomposed trajectory matrix and anti-diagonal averaging to re-construct the new time series from the subgroups. Then it gives algorithm of the SSA technique.

2.2.1 Decomposition

1. First step: Embedding

In the Basic SSA algorithm applied to a one-dimensional time series

Y = (x1, x2, x3,· · · , xN) of length N. We map the time seriesY into the lagged

vectors,Xi = (xi, xi+1, xi+2,· · ·, xi+L−1).

Then build trajectory matrix (Hankel matrix) with equal values on anti-diagonals:

X=       x1 x2 x2 x3 · · · xL · · · xL+1 .. . ... xK xK+1 . .. .._. · · · xN       , (2.1)

WhereLis lag window length which is the single parameter of the embedding, and

K = N−L + 1,, withL <(N+ 1)/2. Embedding is a standard procedure in time series analysis (Hassani, 2007).

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2. Second step: SVD

LetS =XXT,λ1, λ2,· · · , λLare eigenvalues of the matrixS. σ1, σ2,· · ·, σLare the

singular values ofXin the decreasing order andσi=

√

λi. Letd=max{i:λi >0}= rankX,U1, U2,· · ·, Udbe the corresponding left eigenvectors, andVi=XTUi/σi, i=

1,2,· · ·, dbe right singular vectors. Then the SVD of the trajectory matrix can be written as:

X=X1+X2· · ·Xd, (2.2)

whereX = UΣVT =Pd

i=1σiUiViT, i = 1,2,· · · , d. Ui(called ’factor empirical

or-thogonal functions’ or EOFs) andVi (called ’principal components’) stand for the

left and right eigenvectors ofX. The collection (Σi, UiandVi) is calledithX

eigen-triple(abbreviated asET).

2.2.2 Reconstruction

1. First step: Grouping The grouping step corresponds to splitting the elementary matricesXiinto several groups and summing the matrices within each group

(Has-sani, 2007). With a set of indicesI ={i1, i2, . . . , ip}, the matrixXI =Xi1+. . .Xip.

The spilt of the set of indicesJ ={1,2, . . . , d}intomdisjoint subsetsI1, I2, . . . , Im,

m depends on the type of time series. Each will reflect the properties of initial data components which have a meaningful interpretation. The representation is as fol-lows:

X=XI1+. . .XIm. (2.3)

Eigentriple grouping is the procedure to choose the setsI1, I2, . . . , Im.

2. Second step: Anti-diagonal averaging

Each matrixXi,j would be transformed into a new time series from the subgroups

of lengthN at this step. Zis set as the vector as

Z(i) =z₁(i), z(₂i), z₃(i),· · · , z(_Ni), wherekis smaller thanL. Z(i)is corresponds to the matrixXIi. Obtain the series by the averaging along anti-diagonals as:

z_k(i)=                  1 k k P j=1 xj,k−j+1 16k < L 1 L L P j=1 xj,k−j+1 L6k < K+ 1 1 N −k+ 1 N−k+1 P j=k−K+1 xj,k−j+1 K 6k < N (2.4)

Anti-diagonal averaging is applied to produce a reconstructed series

y= (y1, y2, y3,· · ·, yN). In this way, the initial seriesY = (y1, y2, y3,· · · , yN)can be

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The initial time seriesY = (y1, y2, y3,· · ·, yN)can be reconstructed by y_n= m X i=1 z_n(i)(n= 1,2,· · ·, N). (2.5)

2.3 SSA of polar motion time series in

x

direction

Originally, SSA utilized a trajectory matrix of time series in an eigen-spectra decomposi-tion (Vautard and Ghil, 1989). SSA has also been employed to separate between signal and noise component not only for biomedical data but also for economic data (Hassani et al., 2012), estimate time series for signal reconstruction (Golyandina and Stepanov, 2005), in geophysics to study climatic records (Ghil et al., 2002; Chen et al., 2013; Dong et al., 2006; Rangelova et al., 2007 ), do the seismic data denoising and reconstruction ((Oropeza and Sacchi, 2011a)) and polar motion time series analysis (Miller and Malkin, 2013). However, our focus is its application to polar motion time series analysis in the time period spanning from1960to2009both inxdirection and inydirection on separat-ing main components for trend, Chandler wobble, annual wobble and residual.

xp is polar motion time series in x direction. For polar motion time series analysis, in

order to separate the oscillations components (CW and AW), also meanwhile the trend, the most appropriate window lengthLequals12to13years both inxdirection and iny

direction (Cui, 2015). So applying SSA method onxpin these4steps with window length

ofL= 240. We choose the first10modes to analysis at the beginning to separate the main components forxp.

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2.3.1 Decomposition first10modes

Applying SSA method onxptime series. As a result, modes according to singular values are in the decreasing order. So there shows decomposition result of the first 10 modes. The plot is as following:

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In order to separate the reconstructed modes into different components, plotxp ampli-tudes of oscillatory components as following:

FIGURE2.2:xpamplitudes on SSA reconstructed modes

As shown in the figure.2.2, the reconstructed mode1, mode2, mode3and mode4 repre-sent the main oscillatory components. In order to show the amplitude and frequency, use the curve fitting method ’Sum of Sine’ (in cftool of Matlab) to fit the main oscillatory com-ponents in number of terms of 1. The fitting equation is shown as: f =a1sin(b1t+c1)), wherea1 is amplitude,b1 is frequency,c1 is phase andtis time. As a result, we get the

curve fitting parameters as follows:

TABLE2.1: Oscillatory modes’ fitting parameters forxpby SSA

fitting parameter a1[arcsec] b1[rad/year] c1 [rad]

mode 1 0.078 02 5.301 −44.83

mode 2 0.0744 5.304 −39.79

mode 3 0.045 49 6.289 −0.9006

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1. Even if we do not have the priori information about these data, it is possible to conclude that the first four modes reconstructions corresponding to the first four singular values represent the main oscillatory components of the time series.

2. It is easily to figure out from figure.2.2 that both of them are the strongest com-ponents (mostly from0.07 [arcsec]to0.1 [arcsec]) among all Reconstructed mode. Besides, from the table below, it is obvious that reconstructed mode 1 and mode2

forxp have almost the same curve fitting frequency (5.301 [rad/year]for mode 1

and5.304 [rad/year]for mode 2) and amplitude (0.078 02 [arcsec]for mode1and

0.0744 [arcsec]for mode 2). As a result, the reconstructed mode1and mode2are seen as the same component and grouped into one component.

3. xp reconstructed mode 3 and mode 4 are also oscillatory components but with

smaller amplitude compared with mode1 and mode2. The amplitude curves of mode3and mode4are close in 2.2. The amplitudes are from0.04 [arcsec]to 0.06 [arcsec]. As shown in 2.1, the frequency of mode3is6.289 [rad/year]and the same as mode 4 which is higher than mode 1 and mode 2. As a result, reconstructed mode3and mode4are considered as the same oscillatory reconstruction compo-nent but different from the first two.

4. Reconstructed mode 5 for xp in 2.1 is quite different from the rest of the first 10

modes. It is not a periodic signal but a trend. Meanwhile it is quite strong with continuous semi-linear feature, so it should be seen as one main component.

5. Reconstructed mode6forxp is not trend but oscillation. But the amplitude is ob-viously smaller than the mode 1 to 4 and the period is not recognized. So it is neither periodic time series nor trend. So as the other modes. So it is not meaning-ful to study on these reconstructed components. The amplitude is smaller than0.02 [arcsec]corresponding to the small singular values. They are neither periodic time series nor trend. As for the polar motion time series analysis, it is not important to study on these reconstructed components. So we just regard all the left part with-out mode 1 to mode5as residual.

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2.3.2 Reconstruction main components

Group thexpreconstructed mode1and mode2as CW, mode3and mode4as AW, other modes as residual. Reconstructed mode 5 is seen as trend. The main components are shown as:

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CW and AW are reconstructed by grouping modes. In oder to analyze them, we got the curve fitting parameters by curve fitting method with equation 2.3.1.

TABLE2.2: Reconstructed components’ fitting parameters forxpby SSA

fitting parameter a1 [arcsec] b1[rad/year] c1[rad]

CW 0.1524 5.306 −42.39

AW 0.090 57 6.289 −1.684

1. The first sub-figure of figure.2.3 is the original polar motion time series in x di-rection. The second one from reconstructed mode 5 ofxp is semi-linear and non-periodic. It is recognized as trend.

2. The sum reconstructed mode 1 and mode 2 is seen as CW in the polar motion time series analysis. The result is shown in the third sub-figure below. The fitting fre-quency of CW forxp is5.306[Hz]. The equation to calculate the period is:T =

2π f .

So period of CW is1.184 166 [year]which is432.457days. As explained polar Mo-tion time series in 1, the period of CW is about 435 days (Höpfner, 2003). It is almost the similar to the period of CW, so the reconstruction of the CW by SSA is credible.

3. The sum ofxpreconstructed mode 3 and mode 4 are shown in the fourth sub-figure.

It is seen as the AW. Its amplitude by fitting is0.090 57 [arcsec]and period calcu-lated by equation2 is0.999 [year]. As we know, the period of AW is1year and the amplitude is0.1 [arcsec](Höpfner, 2003). So SSA is a feasible method for separating the AW from thexppolar motion time series.

4. Residual ofxpis shown in the last one. It is the sum of the left reconstructed modes whose amplitude is smaller than0.02 [arcsec]. They are not periodic time series or a trend. So we see them as residual.

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2.4 SSA of polar motion time series in

y

direction

yp is the polar motion time series inydirection. Applying SSA methods onypwith the

4steps with the window length of L = 240. As we know that singular value in the de-creasing order, just choose the first10modes to analysis at the beginning to separate the main components.

2.4.1 Decomposition first 10 modes

Applying SSA onyptime series and show the result of the first10modes corresponding

to the singular values. The plot is as following:

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In order to select the main components from the all reconstructed modes, we plot the amplitudes of the oscillatory components. Theyp amplitude of oscillatory modes shows

as following:

FIGURE2.5:ypamplitudes on SSA reconstructed modes

In order to show the amplitude and frequency ofypmain oscillatory components, we use the same curve fitting method of 2.3.1 asxp. As a result, we get the fitting parameters as

following:

TABLE2.3: Oscillatory modes’ fitting parameters forypby SSA

fitting parameter a1[arcsec] b1[rad/year] c1[rad]

mode 2 0.077 752 5.305 −40.17

mode 3 0.073 54 5.303 −41.62

mode 4 0.041 98 6.293 −8.397

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1. It can be easily figure out that the first reconstructed mode with the biggest singular value represents trend.

ypreconstructed mode1in figure.2.4 is semi-linear with a strong feature for

contin-uous growing up. It is easy to differ the mode1from the rest of the first10modes. So reconstructed mode1is seen as the main component for trend. The following four modes reconstruction represent the main oscillatory components ofyptime se-ries.

2. The amplitude curve in figure.2.5 of reconstructed mode2and mode3foryplooks quite similar. It is obvious that they are the strongest components among all re-constructed modes. Fitting amplitude for mode2is0.077 752 [arcsec]and the fre-quency is5.305 [rad/year]. For mode 3, fitting amplitude is0.073 54 [arcsec]and frequency is 5.303 [rad/year]. With almost the same oscillations and amplitude, we see them as one component and the reconstructed mode2and mode3can be grouped into one component.

3. In figure.2.4, reconstructed mode4and mode5 foryp are also oscillatory

compo-nents like mode2and mode3but with smaller amplitudes. As shown in figure.2.3, the fitting amplitude for mode3and mode4are around0.04 [arcsec]and frequency around6.29 [rad/year]. As a result, they are considered as the same reconstruction component.

4. yp reconstructed mode6in figure.2.4 is not trend but oscillation. The amplitude is

obviously smaller than the mode2 to4 and the period is not recognized. So it is neither periodic time series nor trend. As for polar motion time series analysis, it is not important to study on these reconstructed components because of the small amplitude. So as the left reconstructed modes. So we just regard all the left part without mode1to mode5as residual.

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2.4.2 Reconstruction main components

Reconstructed mode1forypis seen as trend. Group the mode2and mode3as CW, mode

4and mode5as AW, other modes as residual. The main components are shown as :

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CW and AW are reconstructed by grouping modes. In oder to analyze them, fit by curve fitting method of 2.3.1 to get the fitting amplitude and frequency.

TABLE2.4: Main reconstructed components forypby SSA

CW 0.151 5.304 −43.99

AW 0.0822 6.29 −2.21

1. Original polar motion time seriesypshows in the first sub-figure of figure.2.6. The second one shows trend which is from reconstructed mode 1 foryp. It is semi-linear

and non-periodic. Compared with figure.2.2 the trend forxp, the singular value for ypis bigger so the slope of the trend is larger and the tendency in the figure is more

obvious. This is verified by the original time series in the first sub-figure.

2. The sum of reconstructed mode 2 and mode 3 for yp is shown in the third

sub-figure. It is seen as CW in the polar motion time series analysis. We get the fitting amplitude and frequency by cftool within the equation of 2.3.1. Within the fitting frequency, calculate the period of CW by 2. Then we get the fitting period result for

1.184 613 [year]which is432.621days.

3. The sum of reconstructed mode 4and mode 5 foryp is shown in the fourth

sub-figure. It is seen as the AW. Its fitting amplitude is around0.822 [arcsec]and period calculated by the equation 2 is0.998 92 [year]. It is an annual oscillation.

4. Residual ofypis shown in the last one of the 2.6. The amplitudes for reconstructed mode6 until the last mode are small. So we sum all the left reconstructed modes without the fist5modes as residual. It is not periodic time series or a trend.

5. As we explain in 1, the period of CW is about435days (Höpfner, 2003). The fitting result of CW which is reconstructed by SSA method is 432.621days. It is almost the same as the period of CW. The period of AW is 1 [year]and the amplitude is

0.1 [arcsec](Höpfner, 2003). The fitting result of AW which is reconstructed by SSA method is 0.998 92 [year]. The periods of AW are almost same. As a result, it is reliable to separate the CW and AW from polar motion time series by SSA method.

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Chapter 3

Multi-channel Singular Spectrum

Analysis

A significant improvement on the application of SSA in 2 arises from its expansion to multiple dimensions, which is called Multi-channel Singular Spectrum Analysis (MSSA) (Oropeza and Sacchi, 2011a). This chapter gives the background with its history and ap-plication for MSSA (Section 3.1), the basic algorithms for MSSA with further details in mathematical equations on the four main steps (Section 3.2), as well as the emphasizing in its application for decomposing polar Motion time series that demonstrate its effec-tiveness in extracting main components (Section 3.3) and some discussions.

3.1 Background

Multi-channel, Multivariate SSA (or MSSA), is a time series analysis method (Read, 1993). It is also known as an Extended Empirical Orthogonal Functions (EEOF) analysis (von Storch and Zwiers, 2002), is a generalization of the SSA over the multidimensional time series (Zotov and Shum, 2010), where the size of different univariate series does not have to be the same. They are both the extensions of classical PCA which is actually a special case of the MSSA when no time lags are introduced (Chen et al., 2013). The trajectory ma-trix of multi-channel time series consists of stacked trajectory matrices of separate times series. The rest of the algorithm is the same as in the univariate case. System of series can be forecasted analogously to SSA recurrent and vector algorithms (Golyandina and Stepanov, 2005).

MSSA provides insight into the unknown or partially known dynamics of the under-lying system by decomposing the delay-coordinate phase space of a given multivariate time series into a set of data-adaptive orthonormal components. These components can be classified essentially into trend, oscillatory patterns and noises (Groth and Ghil, 2011).

It has been applied to intraseasonal variability of large-scale atmospheric fields for pre-dicting climate records (Mo, 2001). Besides, it shows the advantage of reconstruction and denoising on geodetic data (Zotov and Shum, 2010, Oropeza and Sacchi, 2011b and

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Golyandina and Stepanov, 2005) and decomposing the dynamic time series (Groth and Ghil, 2011).

3.2 Methodology

Consider the multi-channel time seriesY =xd(n) :d= 1, . . . , D, n= 1, . . . , N be a

mul-tivariate time series withD channels of lengthN. There are three steps. The first step is embedding original time series Y with lagged vectorXd(i)into the trajectory matrix X. The second step is building covariance matrix to get eigenelements for principal com-ponents. The second step is reconstruction of the time series. Polar motion time series can be separated into the trend, Chandler wobble, annual wobble and residual by MSSA. Then it gives algorithm for MSSA.

3.2.1 Embedding

We map time seriesY =xd(n) :d= 1, . . . , D, n= 1, . . . , Ninto the lagged vectors,Xd(i) =

(xd(i), xd(i+ 1), xd(i+ 2),· · ·, xd(i+L−1)). Then build trajectory matrix (Hankel

ma-trix) which isDM columns of lengthN −L+ 1with equal values on anti-diagonals as following: X=       x1(1) x1(2) x1(2) x1(3) · · · x1(L) · · · x1(L+ 1) .. . ... x1(K) x1(K+ 1) . .. .._. · · · x1(N) . . . xD(1) xD(2) xD(2) xD(3) · · · xD(L) · · · xD(L+ 1) .. . ... xD(K) xD(K+ 1) . .. .._. · · · xD(N)       , (3.1) Where L is lag window length which is the single parameter of the embedding, and

K = N −L + 1,, withL <(N + 1)/2.

3.2.2 Covariance matrix

1. Calculate the grand covariance matrix CX,

CX=

1

N X

T_X. _(3.2)

2. Diagonalized covariance matrix CX,

Λ = QT CXQ. (3.3)

A diagonal matrixΛcontains the real eigenvaluesλkof CX, and a matrixQwhose

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3. Principal components Projecting the trajectoryXonto the eigenvectors,

A=XE, (3.4)

the entries of the PC matrixAcan be written inak,

ak(n) = D X d=1 L X l=1 xd(n+l−1)edk(l), (3.5) withk= 1, . . . , DL, andn= 1, . . . , N−L+ 1. 3.2.3 Reconstruction rdk(n) = 1 Mn Un X l=Ln ak(n−l+ 1)edk(l), (3.6)

whereedk associated eigenvectors at dchannel. The rdk are referred to as

recon-structed components and represent that part of channelxdthat corresponds to the

eigenelement pair(λk, ek). The values of the normalization factorMnand the

sum-mation boundsLnand Unfor the central part of the time series.

3.3 MSSA of bivariate polar motion time series

MSSA method is often applied to multi-components seismic data denoising and the miss-ing values reconstruction ((Oropeza and Sacchi, 2011b)), to study gravity field ((Zotov and Shum, 2010)) and to forecast in the econometric field ((Zhigljavskya et al., 2008)). However, our focus is analyzing polar motion time series analysisxp inxdirection and ypinydirection at the same time in the time period spanning from1960to2009by MSSA and separating main components as trend, CW, AW and residual forxpandyp.

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3.3.1 Separation ofxptime series

Decomposition of first 10 modes

Applying MSSA method onxptime series and plot the result of the first 10 modes corre-sponding to the singular values as following:

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In order to separate the reconstructed modes into different components, plotxp

ampli-tude of main oscillatory components as following:

FIGURE3.2:xpamplitudes on MSSA reconstructed modes

In order to show the amplitude and frequency, we use the same curve fitting method applied in MSSA to show main oscillatory components. As a result, fitting parameters as following:

TABLE3.1: Oscillatory modes’ fitting parameters forxpby MSSA

mode 2 0.077 41 5.306 −43.67

mode 3 0.075 1 5.304 −38.48

mode 4 0.04566 6.289 −1.451

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1. It is obvious that reconstructed both mode1and mode8forxpin figure.3.1 is trend. They are quite different from rest of the first10modes which are oscillation. Mean-while they are quite strong with continuous semi-linear feature, so they should be the same main component forxp.

2. From figure.3.2, we can see that reconstructed mode2, mode3, mode4and mode5

are the main oscillatory components of time seriesxpwith stronger amplitudes. So

we fit the curve with ’cftool’ in Matlab to get fitting parameters shown in table.3.1.

3. As shown in figure.3.2, amplitudes for reconstructed mode2and mode3forxpby

MSSA are almost the same. Fitting frequency for mode2 is5.306 [rad/year]and for mode3is5.306 [rad/year]. As a result, the reconstructed mode2and3can be grouped in one component.

4. xpreconstructed mode4and mode5are also oscillatory components with nearby amplitudes. Their amplitudes are between0.04 [arcsec]and0.06 [arcsec]. As a re-sult, they are considered as the same reconstruction component. Compared with reconstructed mode2 and mode3, the fitting amplitude are smaller and frequen-cies are higher. So reconstructed component for mode4and mode5are seen as the other different oscillatory component from the mode2and mode3.

5. Reconstructed mode6, mode7, mode9and mode10forxpare not trend but

oscil-lation. But the amplitudes are obviously smaller than0.02 [arcsec]and the period is not recognizable. So it is neither periodic time series nor trend. It is not meaning-ful to study on these reconstructed components. So we just regard all the left part without mode1, mode2, mode3, mode4, mode5and mode8as residual.

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Reconstruction main components ofxpby grouping

Group the MSSA reconstructed mode2and mode3forxpas CW, mode4and mode5as AW, other modes as residual. Reconstructed mode1and mode8are seen as trend. The main components are shown as following:

FIGURE3.3: Main reconstructed components forxpby MSSA

CW and AW are reconstructed by grouping modes. In oder to analyze them, we get the fitting parameters for MSSA reconstructed components by curve fitting method of equation.2.3.1.

TABLE3.2: Reconstructed components’ fitting parameters forxpby MSSA

CW 0.1524 5.305 −41.13

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1. The first sub-figure of figure.3.3 is the original time seriesxp. The second one from is semi-linear and non-periodic. It is reconstructed by summing mode 1 and mode 8 and recognized as trend.

2. CW forxpis reconstructed by summing mode 2 and mode 3. The grouping result

is shown in the third sub-figure below. The fitting amplitude for reconstructed CW is0.1524 [arcsec]and frequency is5.305 [rad/year]. As we know frequency, the pe-riod for CW can be calculate byT = 2π

f . So period of reconstructed CW is432.5389

days. As explained polar motion time series in 1, the period of Chandler Wobble is about435days (Höpfner, 2003). It is almost the same as the period of CW, so it is a good idea to reconstruct CW forxpby MSSA .

3. AW is grouped by the sum ofxpreconstructed mode 4 and mode 5 which is shown

in the fourth sub-figure. Its fitting amplitude is 0.090 71 [arcsec]and frequency is

6.289 [rad/year]. So the period for reconstructed AW calculated by equation.2 is

0.999 [year]. As we know, the period of AW is 1 [year]and the amplitude is 0.1 [arcsec](Höpfner, 2003). So MSSA is also a good choice for separating the AW for

xp.

4. Residual ofxpis shown in the last one. It is the sum of the left reconstructed modes whose amplitude is small. They are not periodic time series or a trend. So we see them as residual.

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3.3.2 Separation ofyptime series

Applying MSSA onyptime series and show the result of the first 10 modes correspond-ing to the scorrespond-ingular values. The plot is as followcorrespond-ing:

Decomposition of first 10 modes

Applying MSSA onyptime series and show the result of the first 10 modes corresponding

to the singular values. The plot is as following:

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Amplitude of oscillatory components forypshows as following:

FIGURE3.5:ypamplitudes on MSSA reconstructed modes

As shown in the figure.3.5, reconstructed mode2, mode3, mode4and mode5represent main oscillatory components. In order to get the amplitude and frequency for these os-cillations, we use the same curve fitting method as chapter.2. The fitting parameters is as following:

TABLE3.3: Oscillatory modes’ fitting parameters forypby MSSA

mode 2 0.076 52 5.306 −47.23

mode 3 0.074 39 5.304 −43.09

mode 4 0.0419 6.293 −8.369

mode 5 0.040 24 6.286 5.931

1. The first reconstructed mode according to the biggest singular value is semi-linear. It is easy to differ mode1from the other modes so it is seen as one main component for trend. Reconstructed mode8is also like a trend, but the amplitude is too small to take into account.

2. The following four reconstructed modes represent the main oscillatory components of yp time series. The amplitude curve in figure.3.5 of reconstructed mode 2and

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mode2is0.076 52 [arcsec]and the frequency is5.306 [arcsec]. For mode3is0.074 39 [arcsec]and5.304 [rad/year]. For mode4is0.0419 [arcsec]and6.293 [arcsec]and is0.040 24 [arcsec]and6.286 [arcsec]for mode5. Selecting the modes with similar amplitudes and frequencies, so we group reconstructed mode2with mode 3into one component and mode4and mode5into another reconstruction component.

3. For yp reconstructed mode 6, it is not trend but oscillation. The amplitude is

ob-viously close to 0and the period is not recognized. So it is neither periodic time series nor trend. As for the Polar motion time series analysis, it is not meaning-ful to study on these small amplitudes’ reconstructions. So as the left reconstructed modes. So we just regard all the other modes without mode 1 to mode5as residual.

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Separation main components ofypby grouping

With decomposition modes foryp shown below, we reconstruct mode2 and mode3as CW, mode4and mode5as AW, other modes as residual. Reconstructed mode1is seen as trend. The main components are shown as :

FIGURE3.6: Main reconstructed components forypby MSSA

CW and AW are reconstructed by grouping modes together. Apply curve fitting method to get the fitting amplitude and frequency. Fitting parameters are shown as following:

1. Figure.15 shows the original Polar motion time series in the first sub-figure . Then it is trend ofypreconstructed mode5which is semi-linear and non-periodic.

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TABLE3.4: Reconstructed components’ fitting parameters forypby MSSA

CW 0.1509 5.305 −45.21

AW 0.082 02 6.29 −1.574

2. Grouping yp reconstructed mode 2 and mode 3 as CW. It is shown in the third sub-figure below. Reconstructed CW was fitted by sine function with amplitude of

0.1509 [arcsec]and frequency of5.305 [rad/year]. Then the period calculated by the frequency with the equation.2 is432.5389. As explained in 1, the period of CW is about435days (Höpfner, 2003). Fitting period is almost the same as the research period of CW, so reconstruction of CW forypby MSSA is believable.

3. Besides, grouping yp reconstructed mode3and mode4as AW. It is shown in the

fourth sub-figure. Its amplitude is around0.082 02 [arcsec]and frequency is about

6.29 [rad/year]. So we get the approximate period of AW is0.9989 [year]. With the pre-knowledge, the period of AW is1 [year]and the amplitude is0.1 [arcsec]

(Höpfner, 2003). The reconstructed amplitude and period are close to the reality. so AW reconstruction forypby MSSA is in a good result.

4. Residual ofypis shown at last. It is the sum of the other reconstructed modes

with-out the first5mode. They are not periodic time series or a trend and their amplitude is smaller than0.02 [arcsec]. So we see them as residual foryp.

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Chapter 4

Complex Singular Spectrum Analysis

Complex singular Spectrum Analysis (CSSA) is natural extension of SSA in chapter.2 arises from its expansion to two dimensions with complex value. This chapter gives background for CSSA (Section 4.1), the basic algorithms for CSSA (Section 4.2), as well as the analyzing bivariate time series by combinationsxp+iypin Section 4.3 andyp+ixp

in Section 4.4. At last compare the results from these two kinds of combinations (Section 4.5).

4.1 Background

we can consider one dimensional complex-valued series and apply the complex version of SSA to this one-dimensional series. Complex Singular Spectrum Analysis (CSSA) is a a significant improvement from its expansion to two dimensions of SSA with complex value. It combines bivariate time series together by complex value into one time series. The main algorithm is the same as SSA. CSSA technique expand the SSA technique to the bivariate domain by complex adding.

CSSA uses bivariate time domain data to extract information from short and noisy mul-tivariate time series without prior knowledge of the dynamics affecting the time series. It combines bivariate dimensional data into one dimensional by complex value which is a really novel idea.

Complex values has been used in reconstruction of CW and AW by Fourier basis pursuit band pass filter with combinationxpandyp(Wang et al., 2016). CSSA has been applied to large-scale atmospheric fields for forecasting (Mo, 2001). This thesis tries to use CSSA in geodetic time series analysis.

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4.2 Methodology

CSSA is a method decomposing the original time series into a sum of independent and interpretable components such as a trend, periodic or quasi-periodic components and noise in two direction at the same time. A brief discussion on the methodology of the CSSA technique is shown as following:

4.2.1 Generation Y1 ₌ _x1 1, x12, x13,· · · , x1N andY2 ₌ _x2 1, x22, x23,· · · , x2N

are real-valued time series with length ofN. Firstly generate the new time series asY =Y1+iY2 = (x1, x2, x3,· · ·, xN).

4.2.2 Decomposition

We decompose the time series within two steps. The first step is embedding time series

Y with lagged vector Xi into the trajectory matrix X. The second step singular value

decomposition represents trajectory matrixXas a sum of bi-orthogonal elementary ma-tricesXi.

4.2.3 Reconstruction

There are two separate steps: grouping and anti-diagonal averaging. We reconstruct the original time series by grouping to make subgroups of the decomposed trajectory matrix and anti-diagonal averaging to reconstruct the new time series from the subgroups. All the decomposition and reconstruction algorithms for CSSA are the same as SSA.

4.2.4 Separation

Fourthly, since the general algorithm is written down in real valued form, there is the difference in the form of the SVD performed in the complex-valued space, where the transposition should be Hermitian. The initial time seriesy1 = y₁1, y₂1, y₃1,· · · , y1_N

and y2 = y₁2, y₂2, y2₃,· · ·, y2_Ncan be reconstructed by y1_n= real( m X i=1 z_n(i)(n= 1,2,· · · , N)), (4.1) and y2_n=−imag( m X i=1 z_n(i)(n= 1,2,· · ·, N)). (4.2)

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4.3 CSSA of time series

x

p

+

iy

p

Organize the new time series generating polar motion time seriesxp in the time period spanning from1960to2009for the real part andypfor the imaginary part. Apply CSSA

on the organized time series with the window length ofL = 240. Then we get decom-position for xp based on the real part and yp in the imaginary part of separating main components as trend, Chandler wobble, annual wobble and residual.

4.3.1 Separation of real partxp

Apply CSSA on the new organized complex time series and select the real part of decom-position asxptime series. We analyze the first10modes for real part and separate main components for it. The plot for the mode1to mode10is as following:

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In order to separate modes into different components, we plotxpamplitudes of oscilla-tory modes shows as following:

FIGURE4.2: Real part amplitudes on CSSA reconstructed modes

1. It is obvious to see that real part for the first reconstructed mode represents trend. It is quite different from other modes which are oscillatory reconstructions ofxptime

series.

2. The real part for second mode is oscillatory reconstruction. As shown in figure.4.2, the amplitude is the strongest near by0.15 [arcsec]. So the real part for second re-constructed mode is seen as CW forxp.

3. Real part reconstructed mode3is also oscillatory component but with smaller am-plitude compared with mode2. The amplitude is near by0.1 [arcsec]. As a result, real part of reconstructed mode3is considered as AW forxp.

4. Reconstructed mode4 for real part is also oscillation. But the amplitude is obvi-ously smaller than the mode2and mode3and the period is not easily recognized. So it is neither periodic time series nor trend. So as the other modes. The amplitude is smaller than0.04 [arcsec], So we just regard all the left modes as residual forxp.

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Reconstructed mode1for is seen as trend, mode2as CW, mode3as AW, other modes as residual. Main components are shown as:

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In oder to analyze CW and AW, we use the curve fitting method to get fitting amplitude and frequency.

TABLE4.1: Reconstructed components’ fitting parameters forxpby CSSA

fitting parameter a1[arcsec] b1[rad/year] c1[rad]

CW 0.1517 5.305 −40.88

AW 0.086 38 6.289 −2.565

1. The first sub-figure of figure.4.3 shows the original polar motion time series in x

directionxp. The second one from reconstructed mode1ofxpis trend by CSSA.

2. Reconstructed mode2in the real part is seen as CW forxp and it is shown in the

third sub-figure below. The fitting frequency of reconstructed CW forxp is5.305 [rad/year]. Calculating by equation: T = 2π

f , the period reconstructed CW is

432.5389 days. As explained polar motion time series in chapter.1, the period of CW is about435days (Höpfner, 2003). The result we get is similar to the period of CW, so the reconstruction of the CW forxpby CSSA in the real part is credible.

3. Reconstructed mode3in the real part is shown in the fourth sub-figure. It is seen as the AW forxp. Its amplitude by fitting is0.086 38 [arcsec]and period calculated by equation2 is0.999 [year]. As we know, the period of AW is1year and the ampli-tude is0.1 [arcsec](Höpfner, 2003). So CSSA is a good method for separating the AW from the real part from new reconstructed time series.

4. Residual ofxpis shown in the last one. It is the sum of the left reconstructed modes in the real part whose amplitude is smaller than0.02 [arcsec].

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4.3.2 Separation of imaginary partyp

Apply CSSA on the new organized complex time series and select the imaginary part of decomposition asyp time series. Then plot the first10modes and separate the main

components foryp. The plot for the mode1to mode10is as following:

FIGURE4.4: Reconstructed modes 1–10 of the time series imaginary part by CSSA

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In order to separate the modes into different components by amplitude,ypamplitudes of oscillatory modes shows as following:

FIGURE4.5: Imaginary part amplitudes on CSSA reconstructed modes

1. As shown in figure.4.4, it is obvious to see that the first reconstructed mode for the imaginary part is trend which is quite different from other modes. Others are oscil-latory reconstructions for the imaginary part of time series.

2. The second reconstructed mode for the imaginary part is an oscillation. As shown in 4.5, the amplitude for mode2is the strongest from0.12to0.18 [arcsec]. So it is recognized as CW foryp.

3. Reconstructed mode3forypis also oscillatory component but with smaller ampli-tude compared with mode2. The amplitude is close to0.1 [arcsec]. So we consider reconstructed mode 3 in the imaginary part as AW foryp.

4. The left reconstructed modes for the imaginary part are also oscillations. But ampli-tudes of them is smaller than0.04 [arcsec]and the period is not easily recognized. So it is neither periodic time series nor trend. So we just regard all the left modes as residual foryp.

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Reconstructed mode1 in the imaginary part is seen as trend foryp, mode2 as CW, mode 3 as AW, sum of other modes as residual. Main reconstructed components are shown as following:

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In oder to analyze the new reconstructed components, we get fitting amplitude and fre-quency by ’cftool’ in Matlab. The fitting parameters is as following:

TABLE4.2: Reconstructed components’ fitting parameters forypby CSSA

CW 0.1515 5.305 −45.42

AW 0.086 35 6.29 −1.281

1. Original polar motion time series iny directionyp shows in the first sub-figure of figure.4.6. The second one is trend from reconstructed mode1in the imaginary part forypby CSSA.

2. Reconstructed mode2shown in the third sub-figure below is seen as CW for yp.

The fitting frequency of CW reconstructed by CSSA forypis5.305 [rad/year]. With

equation T = 2π

f , we calculate the period of CW and get the result of 432.5389

days. The period of CW in polar motion time series explained in 1 is about 435

days (Höpfner, 2003). Fitting period of CW reconstructed by CSSA is almost the same to the period of CW, so CSSA is a good choice to reconstruct CW.

3. Reconstructed mode3 in the imaginary part is shown in the fourth sub-figure. It is seen as the AW foryp. Its fitting amplitude is0.086 35 [arcsec]and period calcu-lated by equation.2 with fitting frequency of6.29 [rad/year]is0.9989 [year]. As we know, the period of AW is1year and the amplitude is0.1 [arcsec](Höpfner, 2003). So CSSA is also a method for separating the AW from imaginary part of the new reconstructed time series.

4. The last one is residual foryp. It is the sum of the left reconstructed modes in the imaginary part whose amplitude is smaller than0.04 [arcsec].

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4.4 CSSA of time series

y

p

+

ix

p

Organize a new Polar motion time series iny directionyp for the real part and in x di-rection xp for the imaginary part. Apply CSSA on the organized time series with the

window length ofL= 240. Then we get decomposition forypbased on the real part and xp in the imaginary part of the analysis result to separate main components of the new polar motion time series for trend, Chandler wobble, annual wobble and residual.

4.4.1 Separation of imaginary partxp

Apply CSSA on the new organized complex time series and select imaginary part of decomposition asxptime series. We select the first10modes to analyze and separate the main components forxp. The plot for the mode 1 to mode10is as following:

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In order to separate the modes into different components, amplitudes of oscillatory modes of imaginary part forxpshows as following:

FIGURE4.8: Imaginary part amplitudes on CSSA reconstructed modes

1. Imaginary part of decomposition result isxp. So we see that the first reconstructed mode represents trend forxp. It is quite different from other modes which are

os-cillatory reconstructions.

2. Reconstructed mode2for imaginary part is an oscillatory reconstruction. As shown in figure.4.8, the amplitude is the strongest which is near0.1517 [arcsec]. So the sec-ond reconstructed mode is seen as one main component CW forxp.

3. Reconstructed mode3for imaginary part is also an oscillatory component but with smaller amplitude compared with mode2. The amplitude is near0.08638 [arcsec]. As a result, reconstructed mode3is considered as another main component forxp.

4. Reconstructed mode4is for imaginary part also an oscillation. But the amplitude is obviously smaller than the mode2and mode3and the period is not easily rec-ognized. So it is neither periodic time series nor trend. The other modes also with amplitude which is smaller than0.04 [arcsec], So we just regard all the left modes as residual.

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we generate main components for imaginary partxpare shown as following:

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TABLE4.3: Reconstructed components’ fitting parameters forxpby CSSA

CW 0.1517 5.305 −40.88

AW 0.086 38 6.289 −2.565

1. Original polar motion time series inxdirectionxp shows in the first sub-figure of

figure.4.9. The second one from reconstructed mode1according to the imaginary part of new reconstructed time series is trend by CSSA.

2. Reconstructed mode2in the imaginary part is seen as CW and it is shown in the third sub-figure below. The fitting frequency of CW forxp is5.305 [rad/year]. Use

the equation T = 2π

f to calculate period of CW. So we get the period of CW for

432.5389 days. As explained in chapter.1, the period of CW is about 435 days (Höpfner, 2003). The period for reconstructed CW by CSSA is near to the real pe-riod. So CSSA is a method to separate CW from polar motion time series in the imaginary part.

3. Reconstructed mode3in the imaginary part is shown in the fourth sub-figure. It is seen as the AW forxp. Its amplitude by fitting is0.086 38 [arcsec]and period calcu-lated by equation2 is0.999 [year]. As the period of AW is1year and the amplitude is0.1 [arcsec](Höpfner, 2003). So CSSA is a choice for separating the AW from the complex polar motion time series.

4. Residual forxpis shown in the last one. It is the sum of the left reconstructed modes

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4.4.2 Separation of real partyp

Apply CSSA on the new organized complex time series and select the real part as decom-position foryptime series. We analyze the first10modes in the the real part and separate

the main components for it. The plot for the mode1to mode10is as following:

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In order to separate the modes into different components, amplitudes of oscillatory modes in real part shows as following:

FIGURE4.11: Real part amplitudes on CSSA reconstructed modes

1. It is obvious to see that the first reconstructed mode in the real part represents trend foryp. It is quite different from other modes which are oscillatory reconstructions

in the real part.

2. The second reconstructed mode is an oscillatory reconstruction. As shown in fig-ure.4.11, the amplitude is the strongest from0.1 [arcsec]to0.2 [arcsec]. So the sec-ond reconstructed mode is seen as CW foryp.

3. Reconstructed mode3in the real part is also oscillatory component but with smaller amplitude compared with mode 2. The amplitude is from 0.07 [arcsec] to 0.12 [arcsec]. As a result, reconstructed mode 3 in the real part is considered as AW foryp.

4. Reconstructed mode 4 for yp is also oscillation. But the amplitude is obviously

smaller than the mode2and mode3and the period is not recognizable. So it is nei-ther periodic time series nor trend. So as the onei-ther modes with amplitudes smaller than0.05 [arcsec]. So we just regard all the left modes in the real part as residual

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foryp.

Reconstructed mode1in the real part foryp is seen as trend, mode2as CW, mode 3 as

AW, other modes as residual. Main components forypare shown as:

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TABLE4.4: Reconstructed components’ fitting parameters forypby CSSA

CW 0.1525 5.305 −45.42

AW 0.086 35 6.29 −1.281

1. The first sub-figure of figure.4.12 shows the original Polar motion time series in y directionyp. The second sub-figure reconstructed from mode1 in the real part is trend forypby CSSA.

2. Reconstructed mode2is seen as CW forypand it is shown in the third sub-figure below. The fitting frequency of reconstructed CW forypis5.305 [rad/year].

Calcu-late the period by the equation: T = 2π

f , we get period of CW is432.5389days.

As explained in 1 for polar motion time series, the period of CW is about435days (Höpfner, 2003). It is similar to the period of CW, so the CW reconstruction by CSSA by separating real part is credible.

3. Reconstructed mode3in the real part is shown in the fourth sub-figure. It is seen as AW foryp. Its amplitude by fitting is0.086 35 [arcsec]and period calculated with

fitting frequency is0.9989 [year]. As we know, the period of AW is1year and the amplitude is0.1 [arcsec](Höpfner, 2003). So CSSA is a method for AW reconstruc-tion fromypin the real part.

4. Residual ofypis shown in the last sub-figure. It is the sum of the left reconstructed

modes whose amplitude is smaller than0.05 [arcsec]in the real part.

4.5 Comparison CSSA on

x

p

+

iy

p

with

y

p

+

ix

p

We combine polar motion time seriesxpandypin two ways: xp+iypin Section 4.3 and yp +ixp in Section 4.4. Then apply CSSA on new organized time series and separate main components fromxpandypas trend, CW, AW and residual. In this section, we will

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4.5.1 Comparison ofxpmain components

Apply CSSA on the new organized complex time series in two forms onxp+iyp and

yp+ixpand separatexpinto trend, CW and AW. The plot of the main components’

dif-ferences by the two forms forxpare as following:

FIGURE4.13: Differences of main components forxp

Differences between xp main components separated by CSSA on the two new

recon-structed time series is quite small in the scale of10−15[arcsec]. So we can conclude that it doesn’t matter whetherxpin real part or in imaginary part forxpdecomposition.

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4.5.2 Comparison ofypmain components

Apply CSSA on the new organized complex time series in two forms on xp +iyp and

yp+ixp and separate trend, CW and AW foryp. We compare the decomposition result

for the three main components ofxp+iypwithyp+ixp. The differences show as follow-ing:

FIGURE4.14: Differences of main components foryp

It is obvious that differences between main components forypseparated by CSSA on the two new reconstructed time series is quite small. Differences for CW and trend are in the scale of 10−15[arcsec]and10−16[arcsec]for AW. As a conclusion, it doesn’t matter whetherypin real part or in imaginary part forypdecomposition. CSSA In the next

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Chapter 5

Summary and outlook

This chapter gives comparison MSSA and CSSA with SSA on decomposing polar motion time series (Section 5.1), conclusion (Section 5.2) as well as outlook (Section 5.3).

5.1 Comparison CSSA with SSA and MSSA

5.1.1 Reconstructed modes comparison

This thesis analyzes polar motion observations in the time period spanning from 1960

to2009 both inxdirection andy direction with three methods: SSA, MSSA and CSSA which are nonparametric spectral estimation methods in time series analysis. They are aids in the decomposition of polar motion time series into a sum of independent and in-terpretable components as trend, CW, AW and residual. These components are grouped by different modes according to different singular values. Modes in different methods show in the following:

TABLE5.1: Modes for decomposition main components by different meth-ods methods SSA (forxp) SSA (foryp) MSSA (forxp) MSSA (foryp) CSSA (forxpandyp) Trend mode 5 mode 1 mode 1

and mode 8 mode 1 mode 1 Chandler Wobble mode 1 and mode 2 mode 2 and mode 3 mode 2 and mode 3 mode 2

and mode 3 mode 2 Annual Wobble mode 3 and mode 4 mode 4 and mode 5 mode 4 and mode 5 mode 4

and mode 5 mode 3

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It is easily recognized that there is much difference between different methods for separating main components. For xp trend reconstruction, there is no regular pattern.

But foryp, trend is reconstructed by mode1for all the three methods which means that trend forypis more obvious than forxp.

For CW reconstruction, two modes are grouped in SSA and MSSA methods but just one single mode in CSSA. So as AW reconstruction. But the numbers of modes for AW are bigger than CW which means singular values in SVD for AW are smaller than CW.

For SSA on bothxpandyp, we combine different modes to reconstruct CW and AW. But just use single mode for trend reconstruction.

For MSSA on both xp andyp, we combine mode 2and mode 3to reconstruct CW and

mode4and mode5to AW. But different reconstructed modes for trend reconstruction.

For CSSA on bothxp andyp, we just put single mode to reconstruct trend, CW or AW.

And reconstructed mode for xp and yp are the same. With respect to SSA and MSSA methods, it shows advantage of CSSA that it can analyze bivariate time series at the same time with just single mode for reconstruction.

5.1.2 Main components comparison

SSA, MSSA and CSSA are methods on decomposition ofxp andypinto a sum of trend,

CW, AW and residual. But reconstructions are different by using different methods. Con-sider SSA method as a standard, calculate differences between MSSA and SSA as well as CSSA and SSA.

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comparison forxp components

Decomposition ofxpare different by using different methods. Amplitude differences be-tween CSSA and SSA and CSSA and MSSA are shown as following:

FIGURE5.1: Differences forxp

In oder to evaluate quality of the differences, we calculate the RMS. RMS for thexp

dif-ferences shows as following:

TABLE5.2: RMS for the methods differences forxp

RMS

for differences CW AW trend CSSA-SSA 0.0015 0.0033 0.0024

CSSA-MSSA 8.0749e−04 0.0031 0.0040

The difference between CSSA and SSA forxpCW is in0.0005 [arcsec]with RMS of0.0015.

In addition, difference between CSSA and MSSA for CW is in0.0005 [arcsec]with the RMS of 8.0749e−04. With this values, we can conclude that CW reconstruction with CSSA method is in a high quality.

For xp AW reconstruction, difference between CSSA and SSA is in 0.015 [arcsec]with the RMS of0.0033. And0.01 [arcsec]with the RMS of0.0031between CSSA and MSSA. Although AW reconstruction is not so good as CW reconstruction, CSSA is also a good choice.

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Forxptrend, difference is in0.015 [arcsec]with RMS of0.0024between CSSA and SSA. And0.01 [arcsec]with RMS of0.004between CSSA and MSSA. So CSSA perform well in reconstruction CW, AW and trend forxp.

comparison foryp components

Decomposition ofypare different by using different methods. Amplitude differences are shown as following:

FIGURE5.2: Differences foryp

RMS for the methods differences forypshows as following:

TABLE5.3: RMS for the methods differences foryp

RMS

for differences CW AW trend CSSA-SSA 0.0013 0.0032 2.1443e−04

CSSA-MSSA 7.1005e−04 0.0031 0.0015

ForypCW reconstruction, the difference between CSSA and SSA is in0.0005 [arcsec]with RMS of0.0013. Between CSSA and MSSA for CW is in7.1005e−04 [arcsec]with RMS of

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ForypAW reconstruction, difference between CSSA and SSA is in0.01 [arcsec]with RMS of0.0032. And0.01 [arcsec]with RMS of0.0031between CSSA and MSSA.

Foryptrend reconstruction, difference is in0.0005 [arcsec]with RMS of2.1443e−04

be-tween CSSA and SSA. And0.0005 [arcsec]with RMS of0.0015between CSSA and MSSA. It can conclude that CSSA perform well in decomposing CW, AW and trend fromyp.

5.2 Conclusion

In order to analyze polar motion observations in the time period spanning from1960to

2009both inxdirection andydirection, SSA and MSSA in this thesis is applied to extract CW, AW, trend from polar motion time series.

For new method CSSA, algorithms are straightforward extensions of SSA algorithms to the complex-valued case. With respect to SSA, CSSA can apply on bivariate time series analysis.

New time series xp +iyp or yp +ixp are combination of polar motion time series inx

andydirection. It shows advantage in separating polar motion time series into a slowly varying trend, Chandler wobble, annual wobble and a structureless residual.

Generation of new time series is asxp+iyp oryp+ixp. It doesn’t matter whetherxpin

real part or in imaginary part forxpdecomposition. The same asyp.

With respect to SSA and MSSA, the grouping modes are consistent and unique for com-ponents reconstruction.

5.3 Outlook

Lag window length is the single parameter of embedding to build trajectory matrix. So it is important to select an optimal window length for SSA, MSSA and CSSA. In the future, window length selection for CSSA is worth study.

A joint analysis CSSA is carried out betweenx andy direction. CSSA perform well in decomposition polar motion time series into CW, AW and trend. But as it is bi-variate time series inx andydirection, the next step of research is to find out the usefulness in multi-variate time series analysis.

Areas where SSA and MSSA can be applied are very broad: climatology, marine science, geophysics, engineering, image processing, medicine, econometrics (Golyandina et al.,

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2001). In the future, CSSA can be used in practical applications such as trend extraction, periodicity detection, seasonal adjustment, smoothing and noise reduction.

Complex singular spectrum analysis of earth orientation time series

Geodätisches Institut

Complex Singular Spectrum Analysis

of Earth Orientation Time Series

Masterarbeit im Studiengang

Geodäsie und Geoinformatik

an der Universität Stuttgart

Yang LI

Stuttgart, Juli 2018

Abstract

Contents

List of Abbreviations

Chapter 1

Introduction

1.1

Motivation

1.2

The Earth Orientation Parameters (EOP) time series

1.3

Window length choice of window length on polar motion

series

1.4

Structure of thesis

Chapter 2

Singular Spectrum Analysis

2.1

Background

2.2

Methodology

2.3

SSA of polar motion time series in

x

direction

2.4

SSA of polar motion time series in

y

direction

Chapter 3

Multi-channel Singular Spectrum

Analysis

3.1

Background

3.2

Methodology

3.3

MSSA of bivariate polar motion time series

Chapter 4

Complex Singular Spectrum Analysis

4.1

Background

4.2

Methodology

4.3

CSSA of time series

x

+

iy

4.4

CSSA of time series

y

+

ix

4.5

Comparison CSSA on

x

+

iy

with

y

+

ix

Chapter 5

Summary and outlook

5.1

Comparison CSSA with SSA and MSSA

5.2

Conclusion

5.3

Outlook